Rigorous numerical inclusion of the blow-up time for the Fujita-type equation

Abstract

Multiple studies have addressed the blow-up time of the Fujita-type equation. However, an explicit and sharp inclusion method that tackles this problem is still missing due to several challenging issues. In this paper, we propose a method for obtaining a computable and mathematically rigorous inclusion of the $$L^2(\varOmega )$$ L 2 ( Ω ) blow-up time of a solution to the Fujita-type equation subject to initial and Dirichlet boundary conditions using a numerical verification method. More specifically, we develop a computer-assisted method, by using the numerically verified solution for nonlinear parabolic equations and its estimation of the energy functional, which proves that the concerned solution blows up in the $$L^2(\varOmega )$$ L 2 ( Ω ) sense in finite time with a rigorous estimation of this time. To illustrate how our method actually works, we consider the Fujita-type equation with Dirichlet boundary conditions and the initial function $$u(0,x)=\frac{192}{5}x(x-1)(x^2-x-1)$$ u ( 0 , x ) = 192 5 x ( x - 1 ) ( x 2 - x - 1 ) in a one-dimensional domain $$\varOmega$$ Ω and demonstrate its efficiency in predicting $$L^2(\varOmega )$$ L 2 ( Ω ) blow-up time. The existing theory cannot prove that the solution of the equation blows up in $$L^2(\varOmega )$$ L 2 ( Ω ) . However, our proposed method shows that the solution is the $$L^2(\varOmega )$$ L 2 ( Ω ) blow-up solution and the $$L^2(\varOmega )$$ L 2 ( Ω ) blow-up time is in the interval (0.3068, 0.317713].